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Simplifying 7x2 + 41x + -6 = 8 Reorder the terms: -6 + 41x + 7x2 = 8 Solving -6 + 41x + 7x2 = 8 Solving for variable 'x'. Reorder the terms: -6 + -8 + 41x + 7x2 = 8 + -8 Combine like terms: -6 + -8 = -14 -14 + 41x + 7x2 = 8 + -8 Combine like terms: 8 + -8 = 0 -14 + 41x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2 + 5.857142857x + x2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 5.857142857x + 2 + x2 = 0 + 2 Reorder the terms: -2 + 2 + 5.857142857x + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 5.857142857x + x2 = 0 + 2 5.857142857x + x2 = 0 + 2 Combine like terms: 0 + 2 = 2 5.857142857x + x2 = 2 The x term is 5.857142857x. Take half its coefficient (2.928571429). Square it (8.576530615) and add it to both sides. Add '8.576530615' to each side of the equation. 5.857142857x + 8.576530615 + x2 = 2 + 8.576530615 Reorder the terms: 8.576530615 + 5.857142857x + x2 = 2 + 8.576530615 Combine like terms: 2 + 8.576530615 = 10.576530615 8.576530615 + 5.857142857x + x2 = 10.576530615 Factor a perfect square on the left side: (x + 2.928571429)(x + 2.928571429) = 10.576530615 Calculate the square root of the right side: 3.25215784 Break this problem into two subproblems by setting (x + 2.928571429) equal to 3.25215784 and -3.25215784.Subproblem 1
x + 2.928571429 = 3.25215784 Simplifying x + 2.928571429 = 3.25215784 Reorder the terms: 2.928571429 + x = 3.25215784 Solving 2.928571429 + x = 3.25215784 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.928571429' to each side of the equation. 2.928571429 + -2.928571429 + x = 3.25215784 + -2.928571429 Combine like terms: 2.928571429 + -2.928571429 = 0.000000000 0.000000000 + x = 3.25215784 + -2.928571429 x = 3.25215784 + -2.928571429 Combine like terms: 3.25215784 + -2.928571429 = 0.323586411 x = 0.323586411 Simplifying x = 0.323586411Subproblem 2
x + 2.928571429 = -3.25215784 Simplifying x + 2.928571429 = -3.25215784 Reorder the terms: 2.928571429 + x = -3.25215784 Solving 2.928571429 + x = -3.25215784 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.928571429' to each side of the equation. 2.928571429 + -2.928571429 + x = -3.25215784 + -2.928571429 Combine like terms: 2.928571429 + -2.928571429 = 0.000000000 0.000000000 + x = -3.25215784 + -2.928571429 x = -3.25215784 + -2.928571429 Combine like terms: -3.25215784 + -2.928571429 = -6.180729269 x = -6.180729269 Simplifying x = -6.180729269Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.323586411, -6.180729269}
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